Question

which of the following graphs could represent a cubic function?
a. graph a: a red curve crossing the x-axis, with a zig - zag shape
b. graph b: a vertical red line
c. graph c: a red curve with a single inflection point, crossing the x - axis
d. graph d: a red straight line with a negative slope
options: graph b, graph c, graph a, graph d

Explanation:

Step1: Recall the definition of a function

A function is a relation where each input \(x\) has exactly one output \(y\). This is checked using the vertical - line test. If a vertical line intersects the graph at more than one point, it is not a function.

Step2: Analyze Graph A

Graph A fails the vertical - line test. There are vertical lines that would intersect the graph at more than one point, so it is not a function.

Step3: Analyze Graph B

Graph B is a vertical line. For a vertical line \(x = a\) (where \(a\) is a constant), there are infinitely many \(y\) - values for a single \(x\) - value. By the vertical - line test, it is not a function.

Step4: Analyze Graph C

Graph C passes the vertical - line test. For every \(x\) - value in the domain of the graph, there is exactly one \(y\) - value. A cubic function \(y=ax^{3}+bx^{2}+cx + d\) (\(a
eq0\)) is a polynomial function, and all polynomial functions are functions (pass the vertical - line test). A cubic function has the general shape of a curve that can have up to two turning points, which is consistent with Graph C.

Step5: Analyze Graph D

Graph D is a straight line (a linear function \(y = mx + b\) with \(m
eq0\)), but we are looking for a cubic function. A cubic function is a third - degree polynomial (\(y=ax^{3}+bx^{2}+cx + d,a
eq0\)), and its graph is not a straight line.

Answer:

C. Graph C